3.395 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt{x}} \, dx\)

Optimal. Leaf size=61 \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]

[Out]

2*a^2*c*Sqrt[x] + (2*a*(2*b*c + a*d)*x^(5/2))/5 + (2*b*(b*c + 2*a*d)*x^(9/2))/9
+ (2*b^2*d*x^(13/2))/13

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Rubi [A]  time = 0.0844582, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2))/Sqrt[x],x]

[Out]

2*a^2*c*Sqrt[x] + (2*a*(2*b*c + a*d)*x^(5/2))/5 + (2*b*(b*c + 2*a*d)*x^(9/2))/9
+ (2*b^2*d*x^(13/2))/13

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Rubi in Sympy [A]  time = 12.0753, size = 61, normalized size = 1. \[ 2 a^{2} c \sqrt{x} + \frac{2 a x^{\frac{5}{2}} \left (a d + 2 b c\right )}{5} + \frac{2 b^{2} d x^{\frac{13}{2}}}{13} + \frac{2 b x^{\frac{9}{2}} \left (2 a d + b c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x**(1/2),x)

[Out]

2*a**2*c*sqrt(x) + 2*a*x**(5/2)*(a*d + 2*b*c)/5 + 2*b**2*d*x**(13/2)/13 + 2*b*x*
*(9/2)*(2*a*d + b*c)/9

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Mathematica [A]  time = 0.0313292, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 c+65 b x^4 (2 a d+b c)+117 a x^2 (a d+2 b c)+45 b^2 d x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2))/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(585*a^2*c + 117*a*(2*b*c + a*d)*x^2 + 65*b*(b*c + 2*a*d)*x^4 + 45*b^
2*d*x^6))/585

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{90\,{b}^{2}d{x}^{6}+260\,{x}^{4}abd+130\,{b}^{2}c{x}^{4}+234\,{x}^{2}{a}^{2}d+468\,abc{x}^{2}+1170\,{a}^{2}c}{585}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)/x^(1/2),x)

[Out]

2/585*x^(1/2)*(45*b^2*d*x^6+130*a*b*d*x^4+65*b^2*c*x^4+117*a^2*d*x^2+234*a*b*c*x
^2+585*a^2*c)

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Maxima [A]  time = 1.35791, size = 69, normalized size = 1.13 \[ \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{9}{2}} + 2 \, a^{2} c \sqrt{x} + \frac{2}{5} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/sqrt(x),x, algorithm="maxima")

[Out]

2/13*b^2*d*x^(13/2) + 2/9*(b^2*c + 2*a*b*d)*x^(9/2) + 2*a^2*c*sqrt(x) + 2/5*(2*a
*b*c + a^2*d)*x^(5/2)

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Fricas [A]  time = 0.220174, size = 72, normalized size = 1.18 \[ \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 65 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + 585 \, a^{2} c + 117 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/sqrt(x),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d*x^6 + 65*(b^2*c + 2*a*b*d)*x^4 + 585*a^2*c + 117*(2*a*b*c + a^2*
d)*x^2)*sqrt(x)

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Sympy [A]  time = 6.71769, size = 78, normalized size = 1.28 \[ 2 a^{2} c \sqrt{x} + \frac{2 a^{2} d x^{\frac{5}{2}}}{5} + \frac{4 a b c x^{\frac{5}{2}}}{5} + \frac{4 a b d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)/x**(1/2),x)

[Out]

2*a**2*c*sqrt(x) + 2*a**2*d*x**(5/2)/5 + 4*a*b*c*x**(5/2)/5 + 4*a*b*d*x**(9/2)/9
+ 2*b**2*c*x**(9/2)/9 + 2*b**2*d*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.241127, size = 72, normalized size = 1.18 \[ \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c x^{\frac{9}{2}} + \frac{4}{9} \, a b d x^{\frac{9}{2}} + \frac{4}{5} \, a b c x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d x^{\frac{5}{2}} + 2 \, a^{2} c \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/sqrt(x),x, algorithm="giac")

[Out]

2/13*b^2*d*x^(13/2) + 2/9*b^2*c*x^(9/2) + 4/9*a*b*d*x^(9/2) + 4/5*a*b*c*x^(5/2)
+ 2/5*a^2*d*x^(5/2) + 2*a^2*c*sqrt(x)